On a problem of Erdős

Let s≥2 be an integer. Denote by f ₁(s) the least integer so that every integer l>f ₁(s) is the sum of s distinct primes. Erd?s proved that f ₁(s)<p ₁+p ₂+?+p _s +Cslogs, where p _i is the ith prime and C is an absolute constant. In this paper, we prove that f ₁(s)=p ₁+p ₂+?+p _s +(1+o(1))slogs=p ₂+p ₃+?+p _s+1+o(slogs). This answers a question posed by P. Erd?s.     

Tauberian Theorem of Erdős Revisited

Dedicated to the memory of Paul Erdős In connection with the elementary proof of the prime number theorem, Erdős obtained a striking quadratic Tauberian theorem for sequences. Somewhat later, Siegel indicated in a letter how a powerful "fundamental relation" could be used to simplify the difficult combinatorial proof. Here the author presents his version of the (unpublished) Erdős–Siegel proof. Related Tauberian results by the author are described. Received December 20, 1999     

Variants of the Erdős–Szekeres and Erdős–Hajnal Ramsey problems

Given integers $\ell ,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell }$ with vertex set $v_1<v_2<\cdots <v_n$ and all edges of the form $v_i{v}_j$ where $|i-j|\le \ell$. The Ramsey number $r(P_n^{\ell },P_n^{\ell })$ is the minimum $N$ such that every 2-coloring of $\left(\genfrac{}{}{0ex}{}{\left[N\right]}{2}\right)$ results in a monochromatic copy of $P_n^{\ell }$. It is well-known that $r(P_n^1,P_n^1)={(n-1)}^2+1$. For $\ell >1$, Balko–Cibulka–Král–Kynčl proved that $r(P_n^{\ell },P_n^{\ell })<c_{\ell }{n}^{128\ell }$ and asked for the growth rate for fixed $\ell$. When $\ell =2$, we improve this upper bound substantially by proving $r(P_n^2,P_n^2)<c{n}^{19.5}$. Using this result, we determine the correct tower growth rate of the $k$-uniform hypergraph Ramsey number of a $(k+1)$-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.     

Inverse problems of the Erdős-Ko-Rado type theorems for families of vector spaces and permutations

Science China Mathematics - Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families...     

Theorems of Erdős-Ko-Rado type in geometrical settings

The original Erds-Ko-Rado problem has inspired much research. It started as a study on sets of pairwise intersecting k-subsets in an n-set, then it gave rise to research on sets of pairwise non-trivially intersecting k-dimensional vector spaces in the vector space V (n, q) of dimension n over the finite field of order q, and then research on sets of pairwise non-trivially intersecting generators and planes in finite classical polar spaces. We summarize the main results on the Erds-Ko-Rado problem in these three settings, mention the Erds-Ko-Rado problem in other related settings, and mention open problems for future research.     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

On a problem of Erdős and Rado

We give some improved estimates for the digraph Ramsey numbersr(K _n ^* ,L _m ), the smallest numberp such that any digraph of orderp either has an independent set ofn vertices or contains a transitive tournament of orderm. By results of Baumgartner and of Erdős and Rado, this is equivalent to the following infinite partition problem: for an infinite cardinal κ and positive integersn andm, find the smallest numberp such that

Lattice-point examples for a question of Erdős

Here we establish a set of eight points in general position in the plane, i.e. no three on a line, no four on a circle, and they determine 7 distinct distances, so that, thei-th distance occursi times,i = 1, 2, , 7. The points are embedded in a triangular net, and the distances are not ordered by size or in any other way. We shall show that some known and unknown examples forn < 8 with the above properties may also be lattice points of a similar net.Research (partially) supported by the Hungarian National Foundation for Scientific Research (OTKA) grant, no. 1808.     

On the proof of Erdős’ inequality

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality \({\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}\) for a constrained polynomial p of degree at most n, initially claimed by P. Erd?s, which is different from the one in the paper of T.Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval (?1, 1) and establish a new asymptotically sharp inequality.     

On a problem of Erdős and Graham

In this paper we provide bounds for the size of the solutions of the Diophantine equation

$$\begin{aligned} x(x+1)(x+2)(x+3)(x+m)(x+m+1)(x+m+2)(x+m+3)=y^2, \end{aligned}$$

where \(4\le m\in \mathbb {N}\) is a parameter. We also determine all integral solutions for \(1\le m\le 10^6.\)

     

Erdős–Ko–Rado theorems in finite classical polar spaces

Recently, Erdős–Ko–Rado theorems in finite classical polar spaces have been derived. We present the table with the results of Pepe, Storme and Vanhove on the largest Erdős–Ko–Rado sets of generators in the finite classical polar spaces, and other more recent results by De Boeck, Ihringer and Metsch.     

On a Question of Erdős and Ulam

Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.     

On a bottleneck bipartition conjecture of Erdős

For a graphG, let (U,V)=max{e(U), e(V)} for a bipartition (U, V) ofV(G) withUV=V(G),UV=Ø. Define (G)=min_(U,V ){(U,V)}. Paul Erds conjectures . This paper verifies the conjecture and shows .This work was part of the author's Ph. D. thesis at the University of New Mexico. Research Partially supported by NSA Grant MDA904-92-H-3050.     

In memory of Tibor Gallai Paul Erdős

In memory of Tibor Gallai Paul Erds     

The Erdős-Ko-Rado theorem for finite affine spaces

In [W.N. Hsieh, Intersection theorems for finite vector spaces, Discrete Math. 12 (1975) 1–16], Hsieh obtained the Erd?s-Ko-Rado theorem for finite vector spaces. This paper generalizes Hsieh’s result and obtains the Erd?s-Ko-Rado theorem for finite affine spaces.     

A conjecture of Erdős and Reddy and the rational approximation of Mittag-Leffler type functions

Пусть \(f(z) = \mathop \sum \limits_{k = 0}^\infty a_k z^k ,a_0 \ne 0, a_k \geqq 0 (k \geqq 0)\) — целая функци я,π _n — класс обыкновен ных алгебраических мног очленов степени не вы ше \(n,a \lambda _n (f) = \mathop {\inf }\limits_{p \in \pi _n } \mathop {\sup }\limits_{x \geqq 0} |1/f(x) - 1/p(x)|\) . П. Эрдеш и А. Редди высказали пр едположение, что еслиf(z) имеет порядок ?ε(0, ∞) и $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (f)< 1, TO \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (f) > 0$$ В данной статье показ ано, что для целой функ ции $$E_\omega (z) = \mathop \sum \limits_{n = 0}^\infty \frac{{z^n }}{{\Gamma (1 + n\omega (n))}}$$ , где выполняется $$\lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{{\omega (n)}}{{e + 1}}} \right\}$$ , т.е. $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{1}{{\rho (e + 1)}}} \right\}< 1, a \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) = 0$$ . ФункцияE _ω (z) имеет порядок ?.     

Theorems of Erd?s-Ko-Rado type in polar spaces

We consider Erd?s-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect non-trivially. We characterize the Erd?s-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n+1,q²) with n?2.